What is the recommended way to release the UI from a long running background task in Blazor client-side

You can use Invoke, I modified counter example page to illustrate it, you will need a kind of singleton DI object to avoid running the process for twice.

Remember Blazor is an experimental project. Also, this answer is also an experimental approach.

@page "/counter"

<h1>Counter</h1>

<p>Current count: @currentCount</p>

<button class="btn btn-primary" onclick="@IncrementCount">Click me</button>

@functions {
    int currentCount = 0;

    void IncrementCount()
    {
        currentCount++;
    }

    protected  override void OnInit()
    {
        Invoke(
            //here your task.
            async () =>
            {
                for(var i =0; i< 50; i++)
                {
                    await Task.Delay(1000);
                    currentCount++;
                    StateHasChanged();                    
                    System.Console.WriteLine("Still running ...");
                }
            });
    }
}